Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the specified axis.
27y = x3, y = 0, x = 6
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To find the volume generated by rotating the region bounded by the curves about the x-axis using the method of cylindrical shells, we can follow these steps:
1. First, sketch the region bounded by the given curves.
The given curves are 27y = x^3, y = 0, and x = 6.
Start by drawing the x and y axes, and plot the points where y = 0 and x = 6. Then, plot the curve 27y = x^3.
2. Determine the limits of integration.
Since we are rotating the region about the x-axis, the height of the cylindrical shells will be the difference in y-values between the upper and lower curves. In this case, the lower curve is y = 0 and the upper curve is 27y = x^3.
To find the limits of integration for y, we need to solve the equation 27y = x^3 for y, and then find the y-values where x = 0 and x = 6.
When x = 0:
27y = 0^3
27y = 0
y = 0
When x = 6:
27y = 6^3
27y = 216
y = 8
So, the limits of integration for y are 0 and 8.
3. Setup the integral for the volume using the cylindrical shells method.
The volume of each cylindrical shell can be found using the formula:
dV = 2πrh * dh
Where r is the distance from the axis of rotation (in this case, the x-axis), and h is the height of the shell.
Since we are rotating about the x-axis, the distance from the axis of rotation will be x, and the height of the shell will be the difference in y-values between the curves.
So, the integral for the volume can be set up as follows:
V = ∫[0 to 8] 2π(x) * (27y - 0) dy
4. Evaluate the integral.
Integrate the equation V = 2π ∫[0 to 8] (27xy) dy.
This can be broken down into two separate integrals:
V = 2π ∫[0 to 8] (27xy) dy
= 2π(27x) ∫[0 to 8] y dy
= 2π(27x) [y^2/2] [0 to 8]
Substituting the limits of integration:
V = 2π(27x) [(8^2/2) - (0^2/2)]
= 2π(27x) (32)
= 1728πx
5. Substitute the value of x.
Since we want to find the volume for the entire region, we need to evaluate V for x = 6 (the farthest point).
V = 1728π(6)
= 10368π
Therefore, the volume generated by rotating the region bounded by the given curves about the x-axis is 10368π cubic units.